I usually have Move tool set to select groups and auto-select enabled. I do disable auto-select sometimes because with the move tool active, you can temporarily toggle auto-select by holding down Cmd, which is sometimes better when using the move tool directly.

I use Cmd+Alt+Right Click to select layers in the document. This works with any tool that can toggle to Move tool when pressing Cmd (+Move tool itself). In Windows, you use Ctrl instead of Cmd. Otherwise, it's the same. This method is not tied to the Move tool settings and will always select layers no matter what.

...and if I want to select folders, I Cmd+Left Click. This works assuming Auto-select is enabled in the Move tool settings. Or if I happen to be using the Move tool, I simply Left Click.

The Cmd+Alt+Right Click method has one quirk that can be annoying or awesome depending on the situation. It selects locked layers, whereas the Move toolAuto-select doesn't.

I also want to mention that I often use the Cmd+Alt+Right Click method and the script mentioned here to get its immediate parent.

Very often it's more useful than the Move tool group selection method. Let's say you have this layer structure:

Group 1

Layer

Group 2

Layer

Layer

Group 3

Layer

The Move tool group selection will always give you Group 1, which basically makes it useless if you'd rather select Group 2 or Group 3.

A second workaround could be creating two tool presets. One for selecting layers and one for selecting groups. You could then make two Actions for triggering each tool preset. You should give both Actions shortcut keys.

You should disable Current tool only in the Tool Presets dropdown that in the left side of the toolbar. This way the actions can be triggered with other tools active. This would be kind of annoying because then you have all tool presets visible in this list at all times. Not so annoying if you don't use it regularly. If you don't do that, you can only use these Actions when Move tool is active.

You wouldn't need to disable Current tool only if you could record the activation of Move tool into an Action, but you can't... You could get around that by using a script, because actions can trigger scripts. Then again you could script the whole process: Select Move tool, Select Preset X. Upside with scripts would be that you have more freedom with the type of Shortcut you want to set.